The best root finding algorithm is Newton's method. But check out
https://www.youtube.com/watch?v=I2sjchgXsmk which constructs a function such that Newton's method always get stuck in a cycle of three non roots.
Books on numerical methods mention the importance of having a method that works, even from a bad first guess. They mention Lehmer-Schur and then wimp out because "its too complicated".
This video actually explains it. Clearly. And its not too complicated :-) Or maybe it is, I haven't tried coding it up from the video.
There doesn't seem to be anything here yet